The charging circuit for the key battery is completely separate from the RFID transponder pill stored in the key.

I believe the key remote frequency is 315Mhz, does your spectrum analyzer have sufficient bandwidth and an antenna to receive this frequency? To make things easier, most auto parts stores have car remote battery testers at the counter that will illuminate a light when they see a signal. Those work well to check our keys.

Do you know where in the steering column the 'charger' is located? I don't see anything obvious on realoem.com.

Yes, I'm certain I would have been able to detect any RF coming from the remote portion of the key. Since the odds that all 3 buttons on the key have become non-operational, that's part of my conclusion that the battery is dead even though its nearly brand new.